State Transition Matrix of Relative Motion for the Perturbed Noncircular Reference Orbit

The precise analytic solution with the reference orbit eccentricity and perturbation effects is needed for the relative motion of formation flying satellites. Since Hill’s equations have considerable errors and are insufficient for the long term prediction of the relative motion, the new approach, called geometric method, is proposed to obtain the state transition matrix as a precise solution under the effects due to the reference orbit eccentricity and the gravitational perturbations. Based on the transformation and the state transition matrix for the relative orbital elements, the geometric method gives a precise solution in closed form to 1 st order in J2 for the non-circular reference orbit with mean orbital elements under the existence of the gravitational perturbation J2. Finally, using the transformation matrix from the mean elements to the osculating elements in powers of the eccentricity, the state transition matrix of the relative motion with the osculating elements is derived without solving the differential equations. The results in this paper are based on the J2 effects, but the approach could be extended to include other perturbing forces.